>>Mysql is not started; the green button is dimmed and does not highlight
when hovered. Which is weird because both Stop and Restart buttons are
clickable.

Sir your MySQL is working. Since it has already started you cannot start it again. So the green button is dimmed and not clickable. And when you Stop it, the red button will get dimmed and not clickable then you will be able to Start MySQL again.

>> (In the mean time, is there a shell script that starts Mysql, like the one, apachectl, that starts Apache?)

MySQL doesn't have a script like apachectl but by just typing the following command in Terminal with proper options mysql server can be started easily.

Code

/Applications/AMPPS/mysql/bin/mysqld

For Options/Help:

Code

/Applications/AMPPS/mysql/bin/mysqld --verbose --help

Also you should use --skip-external-locking instead of --skip-locking or use Default Configuration of AMPPS MySQL.